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Put on your thinking caps for a second...

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by , 22nd September 2011 at 05:02 PM (256 Views)
Before I get to the main tooic, I'm curious to see how this new meme-in-progress is coming along (see my last blog). I know at least one of you wanted to jump on board. I wonder if anybody's used it since then (I have at least once). Curious to see how progress is coming along. :D

Anyway, the main topic is this: a thought experiment I came up with years ago. I don't know if anyone else has gotten the same idea or done any studies on it, but I came up with this independently, that's all I know.

So, the question is: Take two parallel lines that stretch as far as you want. Let's just say they're infinite to make things simpler.

Now, put a little place marker on each line.

Next, connect those two points on the lines, with a vertical line. It should look a little something like this.

-------------------O-------------------
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-------------------O-------------------

Got it? Good.

Now, send those two points, still connected to each other by a straight line. in opposite directions on their respective horizontal lines.

Would the line that connects them ever become completely horizontal? Remember that these points can go as far as you want, infinitely. But would the line that connects them (seen as the vertical line above) ever become horizontal as well? Or would it simply keep getting slower and slower and slower in its leaning toward the horizontal, never actually becoming horizontal? Would the line ever get so slow in its progress that it completely stops leaning before it becomes horizontal? There are a lot of precise, increasingly miniscule measurements involved with this.

This question has trumped me ever since it first popped into my head years ago.

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The line would never be horizontal, although it would eventually become so slanted that to all human perception seem horizontal, there is no way it could possibly ever become literally horizontal.
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We can set up your problem like this -> h=heighth of the connecting straight line. x=the horizontal distance between the point where the top line intersects the connecting line and the point where the bottom point intersects the connecting line (at the beginning of your experiment, x is zero and then tends to infinity). h is any positive real number.

Since we're looking at the slope of the connecting line, we can express that as slope=h/x

When x is infinity, the slope is undefined, so unfortunately there is no definate answer to your question.

BUT! If we take the limit, we can look at the slope as x tends towards infinity, and in this case it's 0. So no, the process never becomes so slow that it stops before completely becoming horizontal.

By the way, I'm assuming that the connecting line doesn't have a fixed length, and that the top and bottom horizontal lines are always the same verticle distance from each other.
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Originally Posted by daniedeter
We can set up your problem like this -> h=heighth of the connecting straight line. x=the horizontal distance between the point where the top line intersects the connecting line and the point where the bottom point intersects the connecting line (at the beginning of your experiment, x is zero and then tends to infinity). h is any positive real number.

Since we're looking at the slope of the connecting line, we can express that as slope=h/x

When x is infinity, the slope is undefined, so unfortunately there is no definate answer to your question.

BUT! If we take the limit, we can look at the slope as x tends towards infinity, and in this case it's 0. So no, the process never becomes so slow that it stops before completely becoming horizontal.

By the way, I'm assuming that the connecting line doesn't have a fixed length, and that the top and bottom horizontal lines are always the same verticle distance from each other.
I don't quite understand the math behind it, but it's good that you were able to make some sort of formula to try to explain it. Even if I don't quite get it. :P